# Thread: Another contrived counting problem...

1. ## Another contrived counting problem...

I just don't know how to set this one up. Here it is:

Five married couple stand in a row so that no wife stands next to her own husband. In how many ways can they stand?

Seems like I should throw a derangement in there. But I think using inclusion-exclusion would be easier.

Thanks.

2. Originally Posted by NoFace
Five married couple stand in a row so that no wife stands next to her own husband. In how many ways can they stand?
This is indeed an inclusion-exclusion problem.
Couple I can stand together in $(9!)(2)$ ways.
Couples I&II can stand together in $(8!)\left( {2^2 } \right)$ ways.
Consider the number of ways that at least one couple is together and subtract from the total.
$\sum\limits_{k = 0}^5 {\left( { - 1} \right)^k {5 \choose k}\left[ {(10 - k)!\left( {2^k } \right)} \right]}$.